Almost Everywhere Flatness of a 3-Space with a Loop-Based Wormhole

Abstract

A particular Riemannian metric which originally has been obtained for a well-known coordinate system in the Euclidean 3-space, is shown to specify, in fact, a manifold with boundary. There are two ways to make the manifold complete. One is to identify two halves of the boundary that turns the manifold into Euclidean 3-space as it was done originally. Another is to identify boundaries of two copies of this manifold, that yields a complete manifold which consists of two copies of Euclidean 3-space connected through a round disk. In general relativity this kind of connection is called `loop-based wormhole'. The straightforward calculation of curvature from the metric specified yields an erroneous result, due which the curvature is zero, that is impossible because a manifold with this structure cannot be flat. This paradox is resolved in full correspondence with the generally-accepted definitions.